3.3 Results
3.3.8 Response function diversity increases addressability
types (Figures 3.5EโF). BMP10 alone activated only ACVR1 knockdown cells (Figure 3.5E, word 1). Intermediate levels of BMP9 activated the wild-type cells (word 2), while higher levels additionally activated BMPR2 knockdown cells (word 3). NMuMG and ACVR1 knockdown cells could be simultaneously activated with intermediate levels of both ligands (word 4), while additional BMP9 enabled activation of all three cell types simultaneously (word 5). These results provide additional evidence that the BMP pathway could potentially support addressing.
The ability to achieve addressing in vitro does not demonstrate that addressing occurs in physiological contexts. However, single-cell gene expression atlases reveal that the receptor profiles of NMuMG cells and their perturbed derivatives resemble those in some natural cell types (Table 3.S1) (Tabula Muris Consortium, 2020; Tabula Muris Consortium et al., 2018). It will be interesting to determine whether the profiles analyzed here play natural addressing roles in vivo.
in which each ๐พ๐ ๐ ๐ and ๐๐ ๐ ๐ takes on one of two values (22๐๐ฟ๐๐ด๐๐ต parameter sets, or 65,536). We then simulated the response of each cell type to each ligand word for each biochemical parameter set and computed the mutual information between the sets of ligand words and pathway activities across the library of cell types (Figures 3.6AโB; Methods 3.6.13). Random, rather than grid-based, sampling of ๐พ๐ ๐ ๐ and๐๐ ๐ ๐ produced similar results (Figure 3.S6A). Mutual information values varied broadly across parameter sets, from 0.32 to 1.91 bits, with a median value of 1.36 bits (Figure 3.6B). By refining our search over biochemical parameters, we were able to identify parameters with values as high as 2.38 bits (Methods 3.6.14).
To assess whether mutual information correlates with addressing, we defined an addressability metric, which quantifies how strongly activation patterns differ for different ligand words without requiring specific targeted profiles (Methods 3.6.15).
For every pair of ligand words, we identified the largest fold difference of activation levels across all cell types. This value is high when two ligand words induce distinct responses in at least one cell type. We defined the addressability metric as the lowest such value across all ligand word pairs and calculated this value for a given number of channels ๐ by taking the best choice of all possible subsets of ๐ ligand words.
Using this metric, we analyzed addressability for systems with low, intermediate, and high mutual information (Figure 3.6C). For the parameter set of highest mutual information (2.38 bits), each of the eight ligand words activated a distinct cell type
(B) The distribution of mutual information across biochemical parameters is shown. Dashed lines indicate the lowest (blue), median (cyan), and high- est (green) values. High mutual information indicates that many distinct cell type combinations can be specifically activated by distinct ligand words.
(C) The addressability values of activated subsets are shown for different numbers of channels. The addressability reflects the minimal fold difference in the response of at least one cell type when exposed to any two distinct ligand words (Methods 3.6.15). Results are shown for three sets of biochemical parameters generating the lowest, median, and highest mutual information values.
(D) The parameter set with the lowest mutual information is represented schemat- ically (top), as in Figure 3.2Biii. For these parameters, the responses for the library of 16 cell types are shown as a 4ร4 grid (bottom left). In each response, the๐ฅ- and๐ฆ-axes represent logarithmic titrations of ligands 1 and 2, respectively. All show the same qualitative response of additive (โaโ) behav- ior, differing only in their quantitative sensitivity. Schematically, overlaying four differing responses (highlighted in purple, cyan, red, and green) reveals that different ligand words largely address similar combinations of cell types (bottom right), with relatively few distinct subsets represented.
(E) For the parameter set with the highest mutual information (top), the cell types in the library show a variety of response patterns (bottom left): ratiometric (โrโ), additive (โaโ), imbalance (โiโ), and balance (โbโ), matching the response archetypes (Figure 3.S1) previously observed experimentally (Antebi et al., 2017). One response not fully matching any archetype is unclassified (โuโ).
Schematically, overlaying four differing responses (purple, cyan, red, and green) reveals that different ligand words can address many distinct subsets of cell types (bottom right). Note that complexes tend to have opposite values of affinity and activity parameters as well as other parameter anticorrelations, as analyzed in (GโH).
(F) Violin plots indicate the distribution of mutual information values for systems with different numbers of distinct archetypes represented among individual cell response functions. Note that greater archetype diversity enriches for high mutual information.
(G) Anticorrelation of affinity and activity parameters for the same complex is associated with higher mutual information. We analyzed average properties across bins of 800 parameter sets. To measure the correlation between affinity and activity of complexes, we represented low and high values asโ1 and 1 and computed the dot product between๐พ and๐ vectors. The average correlation and mutual information across bins are plotted.
(H) Parameter sets with high mutual information show anticorrelation in the ac- tivities of complexes with the same receptor but different ligands. Analysis was done analogous to (G).
(I) We defined a fitness function ๐น that rewards parameter sets exhibiting the anticorrelations observed in (GโH).
(J) An evolutionary algorithm identifies parameter sets that maximize๐น. At each iteration, a random parameter value is flipped from low to high or vice versa.
Changes that increase๐น are accepted. Changes that decrease ๐น are accepted with indicated probability (bottom), which depends on a selection pressure parameter๐ . This process is repeated iteratively (Methods 3.6.18).
(K) An evolutionary algorithm enriches for high mutual information. We ran the algorithm with ๐ > 0 to favor anticorrelations or with ๐ = 0 to randomly sample parameters. For each case, we randomly initialized 2,000 parameter sets and performed 200 iterations. We then evaluated the mutual information for the final value of the parameter set and visualized the resulting distribu- tions. Random selection (๐ = 0, blue) led to a similar distribution of values as the systematically sampled parameter sets (cf. Figure 3.6B), while favor- ing anticorrelations (๐ > 0, green) resulted in an overall increase in mutual information.
See also Figure 3.S6.
combination with over 5.5-fold addressability. The median parameter set (1.36 bits) addressed up to seven distinct cell type combinations at an addressability of 1.6, while the parameter set with lowest mutual information (0.32 bits) addressed only three distinct cell type combinations with addressability of 1.2. Overall, a 1-bit difference in mutual information can increase addressing specificity as well as bandwidth, enabling diverse responses to different ligand words.
ing potential. By contrast, the parameter set with the highest mutual information generated a broad diversity of ligand response functions across the cell types, repro- ducing the experimentally observed ratiometric, additive, imbalance detection, and balance detection โarchetypalโ functions (Figure 3.6E; Methods 3.6.16) (Antebi et al., 2017). By generating diverse two-dimensional response functions, this param- eter set allowed each ligand word to activate a distinctive combination of cell types.
In fact, such a correlation between the diversity of response functions and mutual information is seen across the full library of parameter sets (Figure 3.6F).